There are only a couple perched on the rim on that side. El Tovar is really nice - too bad you couldn't get in there. You might have trouble getting into the locations on the rim - when we went, we booked a full year in advance. We stayed in a cabin at Bright Angel and loved it! Our cabin had a fireplace (which we never ended up using) and was the end cabin so we had windows on the front and side. Open the shade and there, in all its glory, was the Canyon! I wasn't overly impressed with the exterior appearance of Kachina Lodge but it is right on the rim. That's essentially it for lodging right on the South Rim. There's at least one other spot within the park that is a ways back from the rim and other locations outside the park.